Koch-profile blow-up conjecture for quintic gKdV

Let $P$ be the solution of the ODE defining the proposed blow-up profile, and let $u(x,t)$ solve the generalized Korteweg–de Vries equation for $n=5$ with smooth initial data $u_0\in L_2(\mathbb{R})$ of sufficiently large mass and with a single maximum. Quintic gKdV blow-up-profile conjecture. There is a blow-up at finite coordinates $(x^*,t^*)$ such that

$$u(x,t)-\frac{1}{L(t)^{2/n}}P\left(\frac{x-x_m(t)}{L(t)}\right)\to\widetilde u\in L_2,$$

where, for $t\sim t^*$,

$$L(t)=C(t^*-t)^{1/3},$$

with $C=C(u_0)$ independent of $t$, and

$$x_m(t)=\gamma L(t)+x^*\qquad\text{as }t\nearrow t^*,$$

where $\gamma=\gamma(u_0)$ is independent of $t$. The conjecture proposes a universal rescaled blow-up profile and corresponding scale and location asymptotics in the quintic case; the paper presents numerical evidence, while the profile is attributed to work cited as Koch.